score:13
We drink champagne on New Year's in large part due to savvy marketing in the late 19th century.
At first, royal favor made champagne an easy sell to the nobility. But with the rise of industrialization in the 19th century, the nobles were no longer guaranteed to be the wealthiest consumers. Champagne producers dangled their products in front of the newly rich merchant class: an aspirational beverage. Of course, these new customers couldn’t afford to drink champagne every day, but they could afford it on special occasions. Soon they began ordering it for all celebrations. Champagne became de rigueur at festivities from weddings to ship christenings -- to ringing in the New Year. (source)
But another aspect of this question is, why champagne and not prosecco or any other sparkling alcoholic beverage? Kolleen Guy, author of When Champagne Became French also cites the savviness of French marketing in the late 19th century, noting that newspaper ads in the 1880s tied together champagne, family togetherness, and occasions like Christmas and New Year's. The French state didn't want this marketing blitz (or champagne's prestige in general) to spill over to other sparkling wines and accidentally benefit Spanish or Italian producers. Accordingly, the government passed domestic legislation and negotiated international agreements that helped to create and protect the identity of "champagne" as something legally distinct from and superior to all other sparkling wines (see pages 1-9 and 120-122 especially).